Ontology

flow "MultivariateStatisticsOntology2" v4.0

# ============================================================
# MULTIVARIATE STATISTICS — ONTOLOGY 2
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)
# ============================================================

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/multivariate-statistics/ontology-2.html"
  title: "Ontology 2"
  domain: "Probability and Statistics"
  category: "Multivariate Statistics"
  entity: MULTIVARIATE_STATISTICS_ONTOLOGY_2
  entity_type: STATISTICAL_ONTOLOGY

DEFINE:
  MULTIVARIATE_STATISTICS_ONTOLOGY_2 :=
    SEMANTIC_MODEL(
      MULTIPLE_RANDOM_VARIABLES,
      JOINT_DISTRIBUTIONS,
      DEPENDENCE_STRUCTURE,
      MATRIX_REPRESENTATION,
      MULTIVARIATE_METHODS
    )

CORE_STRUCTURE:
  RANDOM_VECTOR:
    X := [X_1, X_2, ..., X_p]^T

  OBSERVATION_VECTOR:
    x_i := [x_i1, x_i2, ..., x_ip]^T

  DATA_MATRIX:
    X_data:
      dimensions: n x p
      rows: OBSERVATIONS
      columns: VARIABLES

  MEAN_VECTOR:
    mu := E[X]

  COVARIANCE_MATRIX:
    Sigma :=
      E[
        (X - mu)(X - mu)^T
      ]

ENTITIES:
  OBSERVATION:
    type: MULTIVARIATE_DATA_ENTITY

  VARIABLE:
    type: STATISTICAL_VARIABLE

  RANDOM_VECTOR:
    type: VECTOR_VALUED_RANDOM_VARIABLE

  DATA_MATRIX:
    type: MATRIX

  JOINT_DISTRIBUTION:
    type: MULTIVARIATE_PROBABILITY_DISTRIBUTION

  MARGINAL_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  CONDITIONAL_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  MEAN_VECTOR:
    type: LOCATION_PARAMETER

  COVARIANCE_MATRIX:
    type: DEPENDENCE_MATRIX

  CORRELATION_MATRIX:
    type: STANDARDISED_DEPENDENCE_MATRIX

  PRINCIPAL_COMPONENT:
    type: DERIVED_MULTIVARIATE_VARIABLE

  LATENT_FACTOR:
    type: UNOBSERVED_VARIABLE

  CLASS_LABEL:
    type: CATEGORICAL_TARGET

RELATIONSHIPS:
  OBSERVATIONS
    -> FORM
    -> DATA_MATRIX

  VARIABLES
    -> FORM
    -> RANDOM_VECTOR

  RANDOM_VECTOR
    -> HAS
    -> JOINT_DISTRIBUTION

  RANDOM_VECTOR
    -> HAS
    -> MEAN_VECTOR

  RANDOM_VECTOR
    -> HAS
    -> COVARIANCE_MATRIX

  JOINT_DISTRIBUTION
    -> DEFINES
    -> MARGINAL_DISTRIBUTIONS

  JOINT_DISTRIBUTION
    -> DEFINES
    -> CONDITIONAL_DISTRIBUTIONS

  COVARIANCE_MATRIX
    -> STANDARDISES_TO
    -> CORRELATION_MATRIX

  MULTIVARIATE_METHOD
    -> OPERATES_ON
    -> JOINT_VARIABLE_STRUCTURE

JOINT_PROBABILITY:
  JOINT_DISTRIBUTION:
    f_X(x_1, ..., x_p)

  MARGINALISATION:
    JOINT_DISTRIBUTION
      -> SUM_OR_INTEGRATE_OVER
      -> NON_TARGET_VARIABLES
      -> MARGINAL_DISTRIBUTION

  CONDITIONING:
    JOINT_DISTRIBUTION
      + OBSERVED_COMPONENTS
      -> CONDITIONAL_DISTRIBUTION

DEPENDENCE:
  COVARIANCE:
    Cov(X_i, X_j) :=
      E[
        (X_i - mu_i)
        (X_j - mu_j)
      ]

  CORRELATION:
    Corr(X_i, X_j) :=
      Cov(X_i, X_j)
      /
      sqrt(
        Var(X_i) Var(X_j)
      )

  RULE:
    INDEPENDENCE
      -> ZERO_COVARIANCE
    WHEN:
      REQUIRED_MOMENTS_EXIST

  CAUTION:
    ZERO_COVARIANCE
      !=
    INDEPENDENCE
    IN_GENERAL

SAMPLE_STATISTICS:
  SAMPLE_MEAN:
    x_bar :=
      (1/n)
      * SUM[i=1..n] x_i

  SAMPLE_COVARIANCE:
    S :=
      (1/(n-1))
      * SUM[i=1..n]
        (x_i - x_bar)
        (x_i - x_bar)^T

  SAMPLE_CORRELATION:
    R :=
      STANDARDISE(S)

MULTIVARIATE_NORMAL:
  notation:
    X ~ N_p(mu, Sigma)

  parameters:
    mu := MEAN_VECTOR
    Sigma := COVARIANCE_MATRIX

  require:
    Sigma:
      SYMMETRIC
      POSITIVE_SEMIDEFINITE

  nondegenerate_case:
    Sigma:
      POSITIVE_DEFINITE

  density:
    f(x) :=
      [1 / ((2*pi)^(p/2) * |Sigma|^(1/2))]
      *
      exp(
        -0.5
        * (x-mu)^T
        * Sigma^(-1)
        * (x-mu)
      )

PRINCIPAL_COMPONENT_ANALYSIS:
  alias: PCA

  input:
    CENTERED_OR_STANDARDISED_DATA

  process:
    COMPUTE_COVARIANCE_OR_CORRELATION
      -> EIGEN_DECOMPOSITION
      -> ORDER_EIGENVALUES
      -> PROJECT_DATA

  eigen_relation:
    Sigma v_k :=
      lambda_k v_k

  component:
    PC_k :=
      v_k^T X

  purpose:
    - DIMENSIONALITY_REDUCTION
    - VARIANCE_REPRESENTATION
    - FEATURE_EXTRACTION

FACTOR_ANALYSIS:
  model:
    X :=
      mu
      + Lambda F
      + epsilon

  purpose:
    REPRESENT_SHARED_COVARIANCE
    USING:
      LATENT_FACTORS

  distinction:
    FACTOR_ANALYSIS != PCA

MULTIVARIATE_REGRESSION:
  model:
    Y :=
      X B
      + E

  purpose:
    MODEL_MULTIPLE_RESPONSES
    FROM:
      PREDICTORS

MANOVA:
  full_name:
    MULTIVARIATE_ANALYSIS_OF_VARIANCE

  purpose:
    TEST:
      GROUP_DIFFERENCES
    ACROSS:
      MULTIPLE_DEPENDENT_VARIABLES

DISCRIMINANT_ANALYSIS:
  purpose:
    CLASSIFY:
      OBSERVATIONS
    USING:
      MULTIVARIATE_FEATURES

  methods:
    - LINEAR_DISCRIMINANT_ANALYSIS
    - QUADRATIC_DISCRIMINANT_ANALYSIS

CANONICAL_CORRELATION:
  purpose:
    IDENTIFY:
      MAXIMALLY_CORRELATED_LINEAR_COMBINATIONS
    BETWEEN:
      TWO_VARIABLE_SETS

CLUSTER_ANALYSIS:
  purpose:
    DISCOVER:
      GROUP_STRUCTURE
    FROM:
      MULTIVARIATE_SIMILARITY

  methods:
    - K_MEANS
    - HIERARCHICAL_CLUSTERING
    - MODEL_BASED_CLUSTERING

MAHALANOBIS_DISTANCE:
  definition:
    D_M(x, mu) :=
      sqrt(
        (x-mu)^T
        Sigma^(-1)
        (x-mu)
      )

  requires:
    APPROPRIATE_INVERTIBLE_OR_REGULARISED_COVARIANCE_STRUCTURE

  role:
    SCALE_AND_COVARIANCE_ADJUSTED_DISTANCE

MATRIX_ALGEBRA:
  operations:
    - TRANSPOSE
    - MULTIPLICATION
    - INVERSE
    - DETERMINANT
    - RANK
    - EIGEN_DECOMPOSITION
    - SINGULAR_VALUE_DECOMPOSITION

  role:
    SUPPORT_MULTIVARIATE_COMPUTATION

HIGH_DIMENSIONAL_DATA:
  condition:
    p LARGE_RELATIVE_TO n

  risks:
    - SINGULAR_COVARIANCE
    - OVERFITTING
    - UNSTABLE_PARAMETER_ESTIMATION
    - HIGH_VARIANCE

  responses:
    - REGULARISATION
    - SHRINKAGE
    - SPARSE_MODELLING
    - DIMENSIONALITY_REDUCTION

INFERENCE:
  ESTIMATE:
    - MEAN_VECTOR
    - COVARIANCE_MATRIX
    - CORRELATION_MATRIX
    - MODEL_PARAMETERS
    - LATENT_STRUCTURE

  TEST:
    - MULTIVARIATE_MEANS
    - GROUP_DIFFERENCES
    - COVARIANCE_STRUCTURE
    - DEPENDENCE_STRUCTURE

  PREDICT:
    - MULTIVARIATE_RESPONSES
    - CLASS_MEMBERSHIP
    - FUTURE_OBSERVATIONS

DATA_PIPELINE:
  INPUT:
    RAW_MULTIVARIATE_DATA

  VALIDATE:
    - OBSERVATION_COUNT
    - VARIABLE_COUNT
    - VARIABLE_TYPES
    - MISSINGNESS
    - OUTLIERS
    - VARIABLE_SCALE
    - MATRIX_RANK

  TRANSFORM_OPTIONALLY:
    - CENTER
    - STANDARDISE
    - NORMALISE
    - IMPUTE
    - DIMENSION_REDUCE

  ANALYSE:
    SELECT_MULTIVARIATE_METHOD
      -> VALIDATE_ASSUMPTIONS
      -> FIT_MODEL
      -> EVALUATE_MODEL
      -> INTERPRET_RESULTS

MODEL_CONNECTIONS:
  MULTIVARIATE_STATISTICS_ONTOLOGY_2
    -> CONNECTS_TO
    -> MULTIVARIATE_STATISTICS

  MULTIVARIATE_STATISTICS_ONTOLOGY_2
    -> CONNECTS_TO
    -> COVARIANCE_MATRIX

  MULTIVARIATE_STATISTICS_ONTOLOGY_2
    -> CONNECTS_TO
    -> PRINCIPAL_COMPONENT_ANALYSIS

  MULTIVARIATE_STATISTICS_ONTOLOGY_2
    -> CONNECTS_TO
    -> PROBABILITY_DISTRIBUTIONS

  MULTIVARIATE_STATISTICS_ONTOLOGY_2
    -> CONNECTS_TO
    -> LINEAR_ALGEBRA

  MULTIVARIATE_STATISTICS_ONTOLOGY_2
    -> CONNECTS_TO
    -> MACHINE_LEARNING

  MULTIVARIATE_STATISTICS_ONTOLOGY_2
    -> CONNECTS_TO
    -> BAYESIAN_STATISTICS

APPLICATIONS:
  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> SCIENTIFIC_ANALYSIS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> BIOSTATISTICS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ECONOMETRICS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ENVIRONMENTAL_SCIENCE

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> PATTERN_RECOGNITION

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ARTIFICIAL_INTELLIGENCE

SEMANTIC_EQUIVALENCE:
  "multivariate analysis"
    ~= "joint analysis of multiple variables"

  "random vector"
    ~= "vector-valued random variable"

  "variance-covariance matrix"
    ~= "covariance matrix"

  "principal component analysis"
    ~= "PCA"

  EXCLUDE_EQUIVALENCE:
    MULTIVARIATE_ANALYSIS != INDEPENDENT_UNIVARIATE_ANALYSES
    COVARIANCE != CAUSATION
    CORRELATION != CAUSATION
    PCA != FACTOR_ANALYSIS
    ZERO_COVARIANCE != INDEPENDENCE_BY_DEFAULT

VALIDATION_RULES:
  RULE_1:
    DATA_MATRIX:
      MUST_HAVE:
        CONSISTENT_DIMENSIONS

  RULE_2:
    COVARIANCE_MATRIX:
      MUST_BE:
        SYMMETRIC
        POSITIVE_SEMIDEFINITE

  RULE_3:
    MATRIX_INVERSION:
      REQUIRES:
        APPROPRIATE_RANK
      OTHERWISE:
        USE_APPROPRIATE_REGULARISED_OR_GENERALIZED_METHOD

  RULE_4:
    MODEL_ASSUMPTIONS:
      MUST_BE_EXPLICITLY_CHECKED

  RULE_5:
    VARIABLE_SCALE:
      MUST_BE_CONSIDERED
      BEFORE:
        DISTANCE_OR_VARIANCE_BASED_ANALYSIS

  RULE_6:
    SAMPLE_SIZE:
      MUST_BE_EVALUATED
      RELATIVE_TO:
        DIMENSION
        MODEL_COMPLEXITY

  RULE_7:
    MULTIVARIATE_DEPENDENCE:
      MUST_NOT_BE_REDUCED_TO:
        PAIRWISE_CORRELATION_ALONE

CONFIDENCE:
  ontology_identity: 0.99
  multivariate_structure: 0.99
  covariance_structure: 0.99
  distribution_structure: 0.98
  method_relationships: 0.99
  inference_structure: 0.98

UPDATE_MODEL:
  IF:
    NEW_OBSERVATIONS_ARRIVE

  THEN:
    UPDATE:
      DATA_MATRIX
      SAMPLE_MEAN
      SAMPLE_COVARIANCE
      SAMPLE_CORRELATION

    OPTIONAL_RECOMPUTE:
      PCA
      FACTOR_ANALYSIS
      REGRESSION
      CLASSIFICATION
      CLUSTERING

  IF:
    NEW_VARIABLES_ARRIVE

  THEN:
    EXPAND:
      RANDOM_VECTOR
      DATA_MATRIX
      MEAN_VECTOR
      COVARIANCE_MATRIX

    REVALIDATE:
      MATRIX_RANK
      SAMPLE_SIZE
      MODEL_ASSUMPTIONS

  PRESERVE:
    VARIABLE_IDENTITIES
    VARIABLE_UNITS
    OBSERVATION_IDENTITIES
    DATA_PROVENANCE
    TRANSFORMATION_HISTORY

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_VARIABLES
    - IDENTIFY_OBSERVATIONS
    - CONSTRUCT_DATA_MATRIX
    - VALIDATE_DATA_QUALITY
    - ASSESS_VARIABLE_SCALE
    - COMPUTE_MEAN_AND_DEPENDENCE_STRUCTURE
    - IDENTIFY_DISTRIBUTIONAL_ASSUMPTIONS
    - SELECT_MULTIVARIATE_METHOD
    - VALIDATE_MATRIX_AND_MODEL_ASSUMPTIONS
    - EXECUTE_ANALYSIS
    - INTERPRET_JOINT_STRUCTURE
    - RETURN_UNCERTAINTY_AND_LIMITATIONS

  constraints:
    - PRESERVE_JOINT_VARIABLE_STRUCTURE
    - DO_NOT_INFER_CAUSATION_FROM_ASSOCIATION
    - DO_NOT_ASSUME_MULTIVARIATE_NORMALITY
    - CHECK_MATRIX_RANK_BEFORE_INVERSION
    - EXPOSE_SCALING_DECISIONS
    - DISTINGUISH_PCA_FROM_FACTOR_ANALYSIS

FLOW_HASH:
  canonical_entity: "multivariate-statistics-ontology-2"
  namespace: "probability-statistics/multivariate-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - JOINT_VARIABLE_STRUCTURE
    - COVARIANCE_STRUCTURE
    - MULTIVARIATE_METHOD_RELATIONSHIPS

END_FLOW